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Analysis of top-antitop production and dilepton decay events and the top quark mass
Physics Letters B 287 (1992) 225-230 North-Holland
PHYSIC S LETTERS B
Analysis of top-antitop production and dilepton decay events and the top quark mass R . H . D a l i t z a a n d G a r y R. G o l d s t e i n b
" Departmentof Theoretical Physics, Universityof Oxford, Oxford OX1 3NP, UK b PhysicsDepartment, Tufts University, Medford, MA 02155, USA
Received 22 May 1992
A simple idealised procedure is proposed for the analysis of individual top-antitop quark pair production and dilepton decay events, in terms of the top quark mass. This procedure is illustrated by its application to the CDF candidate event. If this event really represents top-antitop production and decay, then the top quark mass would be 131 _+~ GeV.
It is now known that the top quark t is very heavy [ 1 ], so heavy that its electroweak decay t--,bW + is faster than its hadronization [2,3]. It also appears possible that the Tevatron at F N A L may be sufficiently energetic for the creation of t - t pairs,
where (l, ~') denote the two leptons. Since this final state includes two energetic neutrinos, it might appear a hopeless task for analysis and it is this analysis problem which we wish to illuminate here. Let us consider the decay sequence
for an isolated quark in the laboratory frame. Denote the three-momentum of each particle (say a ) by its own name at and its mass and energy by the corresponding m~ and E~, the mass being taken to be zero for all leptons. We now specify a procedure for characterizing all possible final configurations, illustrated by fig. 1. From an origin P, we lay out the m o m e n t a b and 1 in succession. As an expression o f the two constraints, (i) that the system ( b + W ÷) has mass mr, and (ii) that the system (~÷ +v~ ) has mass Mw, we are led [6] to construct a paraboloid with axis along the vector B L = f, with focus at L and chord ULV given by M 2 / E ~ . This has the property that each possible top m o m e n t u m and energy, consistent with the known vectors b and/-for whatever mass mr, corresponds to one point F on this paraboloid, and vice versa. The vector LF then specifies the neutrino mom e n t u m v~. The top quark energy is measured along the axis of symmetry BL, provided that energy ( E b + E 0 is assigned to the point B. A section of the paraboloid perpendicular to this axis, e.g. the circle M N centred on C, consists of the points for all configurations which have the same energy value Et; the radius of this circle is given by the parabolic equation
(a) t--,bW + ,
r2= 5 / 2
~+p--,X+t+t,
(1)
for some final hadronic systems X, and there is already one candidate event [1,4] ~. The most convenient signal for its decay process t--,bW + is through the W + decay modes W ÷--*~+v~, which give a high energy lepton, e ÷, ~t+ or x +, and account in all for about one-third of the W + decay rate. O f course, these leptonic decay modes each involve the emission o f an energetic neutrino, and, with such a leptonic mode for both W + and W - , the final state reached through ( 1 ) is then X + b + ~ + +v~ + 6 + ~ ' - +9~, ,
(b) W + ~ + v ~ ,
(2)
(3)
~ We have used the energy corrections and reference frame definitions given by Sliwa in ref. [ 5 ].
(Et-Eo) ,
(4)
where Eo is the energy at the lowest point O of the paraboloid, given by
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
225
Volume 287, number 1,2,3 Z'
PHYSICS LETTERS B
Z = (Et -E01
F
x
6 August 1992
m2=m~+2Z' Ax/~b2~ ,
(7)
mo being the value of mt for the origin O. This makes it convenient to introduce coordinates (X', Y, Z ' ) with O X ' flying in the plane o f b and I, and O Y as above. The ellipse formed by the intersection of the plane defined by Z ' with the paraboloid surface defined by eq. (4), is then given by
----
(X'-Z'
tan 0 - s e e 0 tan
OM2/2E~) 2
y2
(A see 0) 2
A2
=l, b _ ~ P-
_b,
where
-i
Fig. 1. Shows the paraboloid surface for all momenta t=PF consistent with the observed b and l, if they result from the decay sequence t-,bW +, W÷ -~+v~. The subset of momenta t consistent with the mass value mt lie on a slanting ellipse (e.g. qdh) whose normal is OZ' and which gives a circle when projected onto the XOYplane (e.g. QDH, here). Otherwise, see ref. [6] and the text above.
Eo=Eb "1-(i M2W'~E + 4E )
(5)
However, the configurations on this circle do not have the same value for mr; the top m o m e n t u m t varies appreciably as F moves around this circle and E 2 t2= m~ is least for the point of this circular section furthermost from P, namely N for the case of fig. 1. We define perpendicular axes O X and O Z as shown in fig. 1, with Z=Et-Eo, the third axis OYbeing parallel to l×b. Our interest is in the case o f one definite value for mr, and our purpose is to deduce this value, as far as is possible, from the data on (b,/). It is readily proved [6] that the points having the same value for mt lie on a slanted plane section of the paraboloid, its normal) O Z ' lying in the plane o f the vectors b a n d / - a n d making an angle 0 (positive if counterclockwise, as shown in fig. 1 ) with the axis BOL, given by tan
O=b±/A,
(6)
where A = (Eb-b.i-) and b± is the component of b perpendicular to L This plane section can be defined by its distance Z ' along this normal, the following relation being deducible from eq. (4.8a) ofref. [6], 226
(8)
(E = Eb+ E,P)
sec0
Z'+~smOtanO
)
.
(9)
Since 0 is independent of rn. these ellipses all have the same eccentricity, but they vary in linear scale as Z ' varies. As mt falls, Z ' falls and the ellipse shrinks until it becomes only the point (labelled m . in fig. 1 ) at which the slanted plane is tangential to the paraboloid. The value mt = m. at this point is the lowest m~ compatible with the observed (b, 7) vectors. It is reached when A = 0. From (9), Z . at this point is
Z.=
--sin0tan0,
(10)
4E~
and eq. (8) then requires further that Y. = O, and that X. = Z . tan 0 + sec 0 tan 0 M2w 2E~ = sec 0 tan 0 ( 2 - s i n 2 0 ) ~
( 11 a)
.
(llb)
At this limiting point, we then have from eqs. (7) and (10),
m2.=m~
M2w b i 2E~ A '
(llc)
leading to the results
(12) and
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PHYSICS LETTERS B
M2w A2= (rntZ--mE) 2E~--~ "
(13)
Since this limiting mass cannot depend on the Lorentz frame used, there is a Lorentz-invariant expression for it,
(M~w+ 2 b b ) rnt >/m. = N/(mzb + 2/:'b)2f.b ,
(14)
given in re[. [ 6 ], where the italics used here denote four-vectors. The inelegant expression (1 lc) given above is identical with this more general result. The (X, Y, Z) coordinates of this tangential point can also be obtained directly, by locating the point at which the parabola Z = (E~/M2w)X2 has slope tan 0, giving the result
M~w
X.=-~tan0,
M~w
Y.=0,
Z.=-~tan20,
(15)
in accord with (10) and ( 1 lb), after rotating the axes by angle 0. In (X, Y, Z) coordinates, the ellipse eq. (8) takes the form X - -~-~-tan0
+y2=A2,
6 August 1992
picture of how the set of configurations for t-decay consistent with the observed (b,//) and an assumed mr, v a r y a s m t varies. For t-i pair production, two paraboloids can be constructed, from (b, l-) for t and from (~,//) for i, both in the same laboratory frame. The CDF candidate event [4,5 ] provides a set of momenta for which it is of interest to apply this analysis. The result is shown in fig. 2, where the laboratory space coordinates are (x, y, z), as used by the experimenters. The axis Oz is along the antiproton beam direction, the axes Ox and Oy being transverse. The two paraboloids have been projected onto the yOz plane since b and//happen both to lie approximately in that plane, and the t-ellipses are then seen edge-on, as is apparent in fig. 2. The t- and t-ellipses shown start with the smallest ellipses, for mr= 115 GeV and increase in steps of 10 GeV up to mr=305 GeV. We recall [6] that the inequality (14) gives a lower limit of 110.0 GeV for the top quark mass. Our interest now turns to relating the ellipses of the top paraboloid from this event with those of the antitop paraboloid from the same event.
(16a)
500"~_~>~
with Z = Z ' sec 0 + X t a n 0,
(16b)
Z' being a linear function of m 2, by eq. (7). This circle (16a) is the projection of the slanting ellipse onto the (X, Y) plane, as shown in fig. 1, where it has been projected onto the plane MFN. Its centre is independent of mr and agrees in (X, Y) location with that for the point mr, given by (15) above; its radius has the simple dependence on m, given in eq. (13). This simple result provides us with the following parametrisation for all the points for a definite m t on the surface of the paraboloid: 2
X= M~, tan 0+ A cos a ,
2E,
Z = X t a n 0 + Z ' sec 0,
Y=A sin ~,
(17)
where the parameter a runs from 0 to 2n. The above remarks, together with the forms (17), give a clear
0
-50~03 0 5 -1000
-500
t (z) GeV
0
Fig. 2. The t- and i-paraboloids calculated for the CDF candidate event, if it is assumed to be an example of t - i production with dilepton decays, are projected onto the laboratory (y, z) plane, showing the t- and t-ellipses at 10 GeV intervals from m, = 115 to 305 GeV. It happens that the t-ellipses are seen edge-on in this projection.
227
Volume 287, n u m b e r 1,2,3
PHYSICS LETTERS B
In the simplest parton model, which is all we wish to consider here, the picture of t - t production collisions is that of the reaction patton + antiparton--, t + t .
( 18 )
In this model, following Feynman's original proposal [ 7 ], the patton has momentum xp and the antiparton has momentum - g p ; no transverse m o m e n t u m is permitted because the event is to be seen in the infinite-momentum frame. In this spirit of simplicity, we can relate the top and antitop transverse momenta by requiring that they sum to zero. For given mt, this picks out a finite number of possibilities. In order to visualize them, we rotate the top-paraboloid by 180 ° around the z-axis of the laboratory frame, and project both paraboloids onto the xOy plane. There will then be a projected antitop ellipse and a projected, rotated top ellipse on this transverse plane. Their intersections give the configurations for which (t)t .... .~-~( - t ) t .....
(19)
that is, those configurations for which the t and transverse momenta sum to zero. For the CDF candidate event, a series of these tand t-ellipses projected onto the transverse plane are shown in fig. 3, for mt values from 115 to 185 GeV. In general, two ellipses can have as many as four
300
o
~fi-et.pses
~
\
~.-"~ - ~
~
~I
~'..
/" 1 I"~1.2~ ....
mt = 115
-100
I
I
-200
-100
f (x) GeV
t
I
0
100
200
Fig. 3. The projections of the t- and t-ellipses for definite mt onto the laboratory (x, y) frame, for selected mt values from 115 to 185 GeV, are show to illustrate the m o v e m e n t of the (t (x), t ( y ) ) solutions (i.e. crossings) as mt varies.
228
6 August 1992
crossings and this is indeed the case here, for values of mt between about 120 and 152 GeV. On the other hand, for values of mt less than about 110.2 GeV or exceeding about 410 GeV, the two ellipses do not intersect. In the intervening intervals for mt the two ellipses have only two intersections. When the final state (2) is reached through mechanisms which do not involve intermediate t and t quarks, there will in general be no solutions (i.e. no crossings with m (bW ÷ ) = m ( b W - ) at all. What is the probability of reaching the observed configuration (bW÷bW - ) when the top quark mass has the value mt? There are two factors which stand out in importance: (i) The top and antitop longitudinal momenta. Fig. 2 shows that there is a strong asymmetry between the t- and t-ellipses. The t-quark has its longitudinal momentum ?~ in the same hemisphere as the incident antiproton, whereas the t-quark longitudinal momentum tz is either small or lies in the same hemisphere as ?z. This implies that the t - t CM frame for the CDF candidate event has a large velocity directed in the same hemisphere as the antiproton. To make this remark quantitative, we consider the t - t pair production to be due to the process (18) and calculate the patton and antiparton momenta, xp and -f~p, for assumed mass rn, and the t- and t-momenta at the crossing points for the t- and t-ellipses deduced for this mass value. We give the results in table 1, for m t = 115 to 155 GeV, which will turn out to be the range of most interest. We note that, for the four crossings A to D which occur, x has a value of order 0.1, whereas :e has a large value, typically in the range of 0.3 to 0.5, larger in some cases, and increasing with rn,. The relative probability for each t- or t-configuration is given by the structure functions F(x) or F(X), the net probability then being the product F(x)F(X). As m, increases, the value required for increases until it exceeds unity, which is not a physical possibility: this point is reached at about 220 GeV for the B crossing point and at about 280 GeV for the D crossing point. The interpretation of the CDF candidate event as an example of t - t production in a ~ p collision appears quite unreasonable for m~>200 GeV. Further, the large x and X values required in the acceptable range for rnt imply that the partons and antipartons involved are dominantly valence quarks
Volume 287, number 1,2,3
PHYSICS LETTERS B
6 August 1992
Table 1 For each mt (GeV), structure function products are given for the A, B, C and D crossings, the (xx, ga) being given as percentages; their sums are then multiplied by the lepton factor (22) to give the relative probability for each mr.
mt (GeV)
115 125 135 155
Structure function product A
B
C
D
(XA,gA)
(XB,XB)
(Xc,~c)
(XD,~D)
3.58 (11,43) 1.44 (13,51) 0.55 (14,60) -
10.80 (10,29) 7.77 (12,31) 4.82 (14,35) 1.21 ( 18,48 )
3.19 (9,47) 0.81 (10,59) -
-
and antiquarks, rather than sea-quarks or gluons. (ii) The lepton momenta E~t(E~ ) in the t (i)-rest frame. It was noted in ref. [ 6 ] that
mtE~, =t. l-=b.l- ½MZw
(20)
(with a similar relation for antiparticles), and that its right-hand side, which we denote by K (/~), may be evaluated directly from laboratory measurements; the lepton energy E~, (E~) is then K/mt (K'/mt). With ( V - A ) forms for the weak interactions W ~ Q Q ' and W ~ v ~ , the lepton energy spectrum from the top-decay sequence t-~bW +, W + - ~ + v ~ , , specified in the top quark rest-frame, then has the following normalised form [ 8,9 ]:
4K(m 2 - m Z - Z K ) P(~, rot)= (mt2 --rnb) 2 2 + M w2 ( m t z +mb2)--2M 4 ' (21) and similarly for P(~, mr) in terms of K'. The empirical values for K a n d / ~ could differ widely, but for the CDF candidate, direct evaluation gives K = 5904 GeV 2 a n d / ( ' = 6019 GeV 2. The net probability that the lepton and the antilepton have the energies E~t and E~ deduced from the laboratory data on these events, is therefore
P(L rnt)e(~, m~) .
(22)
Taking (i) and (ii) together, the net probability for the observed t and t production and decay sequences is e ( e v e n t l m t ) = ~ F(x~)F(g,l)P(~, mt)P(~, m~) , 2
8.27 (8,35) 8.53 (8,35) 6.22 (9,37)
Sums F. F
Leptons P. P
Relative probability
14.4
0.029
0.41
20.7
0.113
2.34
14.7
0.227
2.46
7.4
0.198
1.47
where the sum is over all t- and t-ellipse crossings, A, B, C, and D, for this mass mr. From Bayes' Theorem, the likelihood function for mr, given these data from the CDF candidate event, is then simply
e(mt Idata) = G(mt)P(~, mt)P(~, mr) × ~, F(xa)F(ga),
(24)
2
where G expresses any prior knowledge of mt (e.g. from other experiments bearing on mr). Here we take G=constant, chosen to make f P ( m t l d a t a ) d m t = 1. The resulting likelihood function is plotted versus mt in fig. 4. We must note here that our P(mt] data) is discontinuous at mt values where the number of crossings changes. When two ellipse crossings move towards coincidence with smooth variation of mr, both make a finite contribution to (24) right up to the point where they coincide; beyond this value for mr, both crossings are gone and their finite contributions fall abruptly to zero. This discontinuity arises from the simplifications in our arguments here. In reality, the top quark has a natural lifetime width of order 1 GeV, and so does the W + boson; their effects will smear out such discontinuities. There will also be much larger statistical uncertainties in the determination of the momenta b, l, b, and I. There are also real physics effects such as low-energy gluon emission, which give non-zero net transverse momentum to the t - t quark pair produced [ 10 ]. All these effects will need assessment, in due course. For the present, we refer to the
(23) 229
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PHYSICS LETTERS B
6 August 1992
In conclusion, we wish to acknowledge valuable discussions with Dr. Krzysztof Sliwa about the C D F program of research. We both acknowledge the hospitality of the D e p a r t m e n t of Theoretical Physics of Oxford University, u n d e r Dr. D. Sherrington. One of us ( G . R . G . ) acknowledges the US D e p a r t m e n t of Energy for partial support during this research; the other ( R . H . D . ) acknowledges the research support provided by the SERC of the UK, as well as the hospitality of the Physics D e p a r t m e n t of Tufts U n i v e r sity, Medford ( M A ) , during two visits in the course of this work.
25 2.0 1.5 10 05 /
References I
110
120
130
I
rnt 1/*0
I
I
150 GeV 160
I
170
180
Fig. 4. The likelihood function for the top-quark mass mr, based
on the interpretation of the CDF candidate event as due to t-t pair production, followed by W÷ and W- leptonic decays. simplified curve in fig. 4, where these discontinuities are apparent. The peak value occurs at 131 GeV, one standard deviation giving the limits 120 and 153 GeV, not in disagreement with our earlier estimate [ 6 ] using a more intuitive argument. O f course, we recognize that our discussion of the C D F candidate event as a t - t event is pure speculation. Nevertheless, the data from this event provide c o n v e n i e n t i n p u t for an illustration of the analysis procedure which we wish to apply to events which may represent t - t production a n d decay.
230
[1]K. Sliwa (CDF Collab.), in: Proc. 25th Rencontre de Moriond on Z physics (Les Arcs, March 1990), ed. J. Tran Thanh Van (Editions Fronti~res, Gif-sur-Yvette, 1990) p. 459. [2] I. Bigi, Phys. Lett. B 175 (1986) 233. [3] I. Bigiet al., Phys. Lett. B 181 (1986) 157. [4] CDF CoUab., F. Abe et al., Phys. Rev. Lett. 64 (1990) 147; Lee Pondrom (CDF Collab.), in: Proc. XXVth Intern. Conf. on High energy physics (Singapore, 1990), eds. K.K. Phua and Y. Yamaguchi, Vol. I (World Scientific, Singapore, 1991) p. 144. [ 5 ] K. Sliwa, Search for the top quark at the Fermilab Collider, in: Proc. Fourth heavy flavor Syrup. (Orsay, France, June 1991 ), in press. [6] R.H. Dalitz and G.R. Goldstein, Phys. Rev. D 45 (1992) 1531. [7] R.P. Feynman, Phys. Rev. Lett. 39 (1969) 1415. [8] J.H. Kuhn, Nucl. Phys. B 237 (1984) 77. [9 ] A. Czarnecki, M. Jezabek and J.H. Kuhn, Nucl. Phys. B 351 (1991) 70. [ 10] CDF Collab., F. Abe et al., Phys. Rev. Lett. 64 (1990) 142.
PHYSIC S LETTERS B
Analysis of top-antitop production and dilepton decay events and the top quark mass R . H . D a l i t z a a n d G a r y R. G o l d s t e i n b
" Departmentof Theoretical Physics, Universityof Oxford, Oxford OX1 3NP, UK b PhysicsDepartment, Tufts University, Medford, MA 02155, USA
Received 22 May 1992
A simple idealised procedure is proposed for the analysis of individual top-antitop quark pair production and dilepton decay events, in terms of the top quark mass. This procedure is illustrated by its application to the CDF candidate event. If this event really represents top-antitop production and decay, then the top quark mass would be 131 _+~ GeV.
It is now known that the top quark t is very heavy [ 1 ], so heavy that its electroweak decay t--,bW + is faster than its hadronization [2,3]. It also appears possible that the Tevatron at F N A L may be sufficiently energetic for the creation of t - t pairs,
where (l, ~') denote the two leptons. Since this final state includes two energetic neutrinos, it might appear a hopeless task for analysis and it is this analysis problem which we wish to illuminate here. Let us consider the decay sequence
for an isolated quark in the laboratory frame. Denote the three-momentum of each particle (say a ) by its own name at and its mass and energy by the corresponding m~ and E~, the mass being taken to be zero for all leptons. We now specify a procedure for characterizing all possible final configurations, illustrated by fig. 1. From an origin P, we lay out the m o m e n t a b and 1 in succession. As an expression o f the two constraints, (i) that the system ( b + W ÷) has mass mr, and (ii) that the system (~÷ +v~ ) has mass Mw, we are led [6] to construct a paraboloid with axis along the vector B L = f, with focus at L and chord ULV given by M 2 / E ~ . This has the property that each possible top m o m e n t u m and energy, consistent with the known vectors b and/-for whatever mass mr, corresponds to one point F on this paraboloid, and vice versa. The vector LF then specifies the neutrino mom e n t u m v~. The top quark energy is measured along the axis of symmetry BL, provided that energy ( E b + E 0 is assigned to the point B. A section of the paraboloid perpendicular to this axis, e.g. the circle M N centred on C, consists of the points for all configurations which have the same energy value Et; the radius of this circle is given by the parabolic equation
(a) t--,bW + ,
r2= 5 / 2
~+p--,X+t+t,
(1)
for some final hadronic systems X, and there is already one candidate event [1,4] ~. The most convenient signal for its decay process t--,bW + is through the W + decay modes W ÷--*~+v~, which give a high energy lepton, e ÷, ~t+ or x +, and account in all for about one-third of the W + decay rate. O f course, these leptonic decay modes each involve the emission o f an energetic neutrino, and, with such a leptonic mode for both W + and W - , the final state reached through ( 1 ) is then X + b + ~ + +v~ + 6 + ~ ' - +9~, ,
(b) W + ~ + v ~ ,
(2)
(3)
~ We have used the energy corrections and reference frame definitions given by Sliwa in ref. [ 5 ].
(Et-Eo) ,
(4)
where Eo is the energy at the lowest point O of the paraboloid, given by
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
225
Volume 287, number 1,2,3 Z'
PHYSICS LETTERS B
Z = (Et -E01
F
x
6 August 1992
m2=m~+2Z' Ax/~b2~ ,
(7)
mo being the value of mt for the origin O. This makes it convenient to introduce coordinates (X', Y, Z ' ) with O X ' flying in the plane o f b and I, and O Y as above. The ellipse formed by the intersection of the plane defined by Z ' with the paraboloid surface defined by eq. (4), is then given by
----
(X'-Z'
tan 0 - s e e 0 tan
OM2/2E~) 2
y2
(A see 0) 2
A2
=l, b _ ~ P-
_b,
where
-i
Fig. 1. Shows the paraboloid surface for all momenta t=PF consistent with the observed b and l, if they result from the decay sequence t-,bW +, W÷ -~+v~. The subset of momenta t consistent with the mass value mt lie on a slanting ellipse (e.g. qdh) whose normal is OZ' and which gives a circle when projected onto the XOYplane (e.g. QDH, here). Otherwise, see ref. [6] and the text above.
Eo=Eb "1-(i M2W'~E + 4E )
(5)
However, the configurations on this circle do not have the same value for mr; the top m o m e n t u m t varies appreciably as F moves around this circle and E 2 t2= m~ is least for the point of this circular section furthermost from P, namely N for the case of fig. 1. We define perpendicular axes O X and O Z as shown in fig. 1, with Z=Et-Eo, the third axis OYbeing parallel to l×b. Our interest is in the case o f one definite value for mr, and our purpose is to deduce this value, as far as is possible, from the data on (b,/). It is readily proved [6] that the points having the same value for mt lie on a slanted plane section of the paraboloid, its normal) O Z ' lying in the plane o f the vectors b a n d / - a n d making an angle 0 (positive if counterclockwise, as shown in fig. 1 ) with the axis BOL, given by tan
O=b±/A,
(6)
where A = (Eb-b.i-) and b± is the component of b perpendicular to L This plane section can be defined by its distance Z ' along this normal, the following relation being deducible from eq. (4.8a) ofref. [6], 226
(8)
(E = Eb+ E,P)
sec0
Z'+~smOtanO
)
.
(9)
Since 0 is independent of rn. these ellipses all have the same eccentricity, but they vary in linear scale as Z ' varies. As mt falls, Z ' falls and the ellipse shrinks until it becomes only the point (labelled m . in fig. 1 ) at which the slanted plane is tangential to the paraboloid. The value mt = m. at this point is the lowest m~ compatible with the observed (b, 7) vectors. It is reached when A = 0. From (9), Z . at this point is
Z.=
--sin0tan0,
(10)
4E~
and eq. (8) then requires further that Y. = O, and that X. = Z . tan 0 + sec 0 tan 0 M2w 2E~ = sec 0 tan 0 ( 2 - s i n 2 0 ) ~
( 11 a)
.
(llb)
At this limiting point, we then have from eqs. (7) and (10),
m2.=m~
M2w b i 2E~ A '
(llc)
leading to the results
(12) and
Volume 287, number 1,2,3
PHYSICS LETTERS B
M2w A2= (rntZ--mE) 2E~--~ "
(13)
Since this limiting mass cannot depend on the Lorentz frame used, there is a Lorentz-invariant expression for it,
(M~w+ 2 b b ) rnt >/m. = N/(mzb + 2/:'b)2f.b ,
(14)
given in re[. [ 6 ], where the italics used here denote four-vectors. The inelegant expression (1 lc) given above is identical with this more general result. The (X, Y, Z) coordinates of this tangential point can also be obtained directly, by locating the point at which the parabola Z = (E~/M2w)X2 has slope tan 0, giving the result
M~w
X.=-~tan0,
M~w
Y.=0,
Z.=-~tan20,
(15)
in accord with (10) and ( 1 lb), after rotating the axes by angle 0. In (X, Y, Z) coordinates, the ellipse eq. (8) takes the form X - -~-~-tan0
+y2=A2,
6 August 1992
picture of how the set of configurations for t-decay consistent with the observed (b,//) and an assumed mr, v a r y a s m t varies. For t-i pair production, two paraboloids can be constructed, from (b, l-) for t and from (~,//) for i, both in the same laboratory frame. The CDF candidate event [4,5 ] provides a set of momenta for which it is of interest to apply this analysis. The result is shown in fig. 2, where the laboratory space coordinates are (x, y, z), as used by the experimenters. The axis Oz is along the antiproton beam direction, the axes Ox and Oy being transverse. The two paraboloids have been projected onto the yOz plane since b and//happen both to lie approximately in that plane, and the t-ellipses are then seen edge-on, as is apparent in fig. 2. The t- and t-ellipses shown start with the smallest ellipses, for mr= 115 GeV and increase in steps of 10 GeV up to mr=305 GeV. We recall [6] that the inequality (14) gives a lower limit of 110.0 GeV for the top quark mass. Our interest now turns to relating the ellipses of the top paraboloid from this event with those of the antitop paraboloid from the same event.
(16a)
500"~_~>~
with Z = Z ' sec 0 + X t a n 0,
(16b)
Z' being a linear function of m 2, by eq. (7). This circle (16a) is the projection of the slanting ellipse onto the (X, Y) plane, as shown in fig. 1, where it has been projected onto the plane MFN. Its centre is independent of mr and agrees in (X, Y) location with that for the point mr, given by (15) above; its radius has the simple dependence on m, given in eq. (13). This simple result provides us with the following parametrisation for all the points for a definite m t on the surface of the paraboloid: 2
X= M~, tan 0+ A cos a ,
2E,
Z = X t a n 0 + Z ' sec 0,
Y=A sin ~,
(17)
where the parameter a runs from 0 to 2n. The above remarks, together with the forms (17), give a clear
0
-50~03 0 5 -1000
-500
t (z) GeV
0
Fig. 2. The t- and i-paraboloids calculated for the CDF candidate event, if it is assumed to be an example of t - i production with dilepton decays, are projected onto the laboratory (y, z) plane, showing the t- and t-ellipses at 10 GeV intervals from m, = 115 to 305 GeV. It happens that the t-ellipses are seen edge-on in this projection.
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In the simplest parton model, which is all we wish to consider here, the picture of t - t production collisions is that of the reaction patton + antiparton--, t + t .
( 18 )
In this model, following Feynman's original proposal [ 7 ], the patton has momentum xp and the antiparton has momentum - g p ; no transverse m o m e n t u m is permitted because the event is to be seen in the infinite-momentum frame. In this spirit of simplicity, we can relate the top and antitop transverse momenta by requiring that they sum to zero. For given mt, this picks out a finite number of possibilities. In order to visualize them, we rotate the top-paraboloid by 180 ° around the z-axis of the laboratory frame, and project both paraboloids onto the xOy plane. There will then be a projected antitop ellipse and a projected, rotated top ellipse on this transverse plane. Their intersections give the configurations for which (t)t .... .~-~( - t ) t .....
(19)
that is, those configurations for which the t and transverse momenta sum to zero. For the CDF candidate event, a series of these tand t-ellipses projected onto the transverse plane are shown in fig. 3, for mt values from 115 to 185 GeV. In general, two ellipses can have as many as four
300
o
~fi-et.pses
~
\
~.-"~ - ~
~
~I
~'..
/" 1 I"~1.2~ ....
mt = 115
-100
I
I
-200
-100
f (x) GeV
t
I
0
100
200
Fig. 3. The projections of the t- and t-ellipses for definite mt onto the laboratory (x, y) frame, for selected mt values from 115 to 185 GeV, are show to illustrate the m o v e m e n t of the (t (x), t ( y ) ) solutions (i.e. crossings) as mt varies.
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6 August 1992
crossings and this is indeed the case here, for values of mt between about 120 and 152 GeV. On the other hand, for values of mt less than about 110.2 GeV or exceeding about 410 GeV, the two ellipses do not intersect. In the intervening intervals for mt the two ellipses have only two intersections. When the final state (2) is reached through mechanisms which do not involve intermediate t and t quarks, there will in general be no solutions (i.e. no crossings with m (bW ÷ ) = m ( b W - ) at all. What is the probability of reaching the observed configuration (bW÷bW - ) when the top quark mass has the value mt? There are two factors which stand out in importance: (i) The top and antitop longitudinal momenta. Fig. 2 shows that there is a strong asymmetry between the t- and t-ellipses. The t-quark has its longitudinal momentum ?~ in the same hemisphere as the incident antiproton, whereas the t-quark longitudinal momentum tz is either small or lies in the same hemisphere as ?z. This implies that the t - t CM frame for the CDF candidate event has a large velocity directed in the same hemisphere as the antiproton. To make this remark quantitative, we consider the t - t pair production to be due to the process (18) and calculate the patton and antiparton momenta, xp and -f~p, for assumed mass rn, and the t- and t-momenta at the crossing points for the t- and t-ellipses deduced for this mass value. We give the results in table 1, for m t = 115 to 155 GeV, which will turn out to be the range of most interest. We note that, for the four crossings A to D which occur, x has a value of order 0.1, whereas :e has a large value, typically in the range of 0.3 to 0.5, larger in some cases, and increasing with rn,. The relative probability for each t- or t-configuration is given by the structure functions F(x) or F(X), the net probability then being the product F(x)F(X). As m, increases, the value required for increases until it exceeds unity, which is not a physical possibility: this point is reached at about 220 GeV for the B crossing point and at about 280 GeV for the D crossing point. The interpretation of the CDF candidate event as an example of t - t production in a ~ p collision appears quite unreasonable for m~>200 GeV. Further, the large x and X values required in the acceptable range for rnt imply that the partons and antipartons involved are dominantly valence quarks
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Table 1 For each mt (GeV), structure function products are given for the A, B, C and D crossings, the (xx, ga) being given as percentages; their sums are then multiplied by the lepton factor (22) to give the relative probability for each mr.
mt (GeV)
115 125 135 155
Structure function product A
B
C
D
(XA,gA)
(XB,XB)
(Xc,~c)
(XD,~D)
3.58 (11,43) 1.44 (13,51) 0.55 (14,60) -
10.80 (10,29) 7.77 (12,31) 4.82 (14,35) 1.21 ( 18,48 )
3.19 (9,47) 0.81 (10,59) -
-
and antiquarks, rather than sea-quarks or gluons. (ii) The lepton momenta E~t(E~ ) in the t (i)-rest frame. It was noted in ref. [ 6 ] that
mtE~, =t. l-=b.l- ½MZw
(20)
(with a similar relation for antiparticles), and that its right-hand side, which we denote by K (/~), may be evaluated directly from laboratory measurements; the lepton energy E~, (E~) is then K/mt (K'/mt). With ( V - A ) forms for the weak interactions W ~ Q Q ' and W ~ v ~ , the lepton energy spectrum from the top-decay sequence t-~bW +, W + - ~ + v ~ , , specified in the top quark rest-frame, then has the following normalised form [ 8,9 ]:
4K(m 2 - m Z - Z K ) P(~, rot)= (mt2 --rnb) 2 2 + M w2 ( m t z +mb2)--2M 4 ' (21) and similarly for P(~, mr) in terms of K'. The empirical values for K a n d / ~ could differ widely, but for the CDF candidate, direct evaluation gives K = 5904 GeV 2 a n d / ( ' = 6019 GeV 2. The net probability that the lepton and the antilepton have the energies E~t and E~ deduced from the laboratory data on these events, is therefore
P(L rnt)e(~, m~) .
(22)
Taking (i) and (ii) together, the net probability for the observed t and t production and decay sequences is e ( e v e n t l m t ) = ~ F(x~)F(g,l)P(~, mt)P(~, m~) , 2
8.27 (8,35) 8.53 (8,35) 6.22 (9,37)
Sums F. F
Leptons P. P
Relative probability
14.4
0.029
0.41
20.7
0.113
2.34
14.7
0.227
2.46
7.4
0.198
1.47
where the sum is over all t- and t-ellipse crossings, A, B, C, and D, for this mass mr. From Bayes' Theorem, the likelihood function for mr, given these data from the CDF candidate event, is then simply
e(mt Idata) = G(mt)P(~, mt)P(~, mr) × ~, F(xa)F(ga),
(24)
2
where G expresses any prior knowledge of mt (e.g. from other experiments bearing on mr). Here we take G=constant, chosen to make f P ( m t l d a t a ) d m t = 1. The resulting likelihood function is plotted versus mt in fig. 4. We must note here that our P(mt] data) is discontinuous at mt values where the number of crossings changes. When two ellipse crossings move towards coincidence with smooth variation of mr, both make a finite contribution to (24) right up to the point where they coincide; beyond this value for mr, both crossings are gone and their finite contributions fall abruptly to zero. This discontinuity arises from the simplifications in our arguments here. In reality, the top quark has a natural lifetime width of order 1 GeV, and so does the W + boson; their effects will smear out such discontinuities. There will also be much larger statistical uncertainties in the determination of the momenta b, l, b, and I. There are also real physics effects such as low-energy gluon emission, which give non-zero net transverse momentum to the t - t quark pair produced [ 10 ]. All these effects will need assessment, in due course. For the present, we refer to the
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In conclusion, we wish to acknowledge valuable discussions with Dr. Krzysztof Sliwa about the C D F program of research. We both acknowledge the hospitality of the D e p a r t m e n t of Theoretical Physics of Oxford University, u n d e r Dr. D. Sherrington. One of us ( G . R . G . ) acknowledges the US D e p a r t m e n t of Energy for partial support during this research; the other ( R . H . D . ) acknowledges the research support provided by the SERC of the UK, as well as the hospitality of the Physics D e p a r t m e n t of Tufts U n i v e r sity, Medford ( M A ) , during two visits in the course of this work.
25 2.0 1.5 10 05 /
References I
110
120
130
I
rnt 1/*0
I
I
150 GeV 160
I
170
180
Fig. 4. The likelihood function for the top-quark mass mr, based
on the interpretation of the CDF candidate event as due to t-t pair production, followed by W÷ and W- leptonic decays. simplified curve in fig. 4, where these discontinuities are apparent. The peak value occurs at 131 GeV, one standard deviation giving the limits 120 and 153 GeV, not in disagreement with our earlier estimate [ 6 ] using a more intuitive argument. O f course, we recognize that our discussion of the C D F candidate event as a t - t event is pure speculation. Nevertheless, the data from this event provide c o n v e n i e n t i n p u t for an illustration of the analysis procedure which we wish to apply to events which may represent t - t production a n d decay.
230
[1]K. Sliwa (CDF Collab.), in: Proc. 25th Rencontre de Moriond on Z physics (Les Arcs, March 1990), ed. J. Tran Thanh Van (Editions Fronti~res, Gif-sur-Yvette, 1990) p. 459. [2] I. Bigi, Phys. Lett. B 175 (1986) 233. [3] I. Bigiet al., Phys. Lett. B 181 (1986) 157. [4] CDF CoUab., F. Abe et al., Phys. Rev. Lett. 64 (1990) 147; Lee Pondrom (CDF Collab.), in: Proc. XXVth Intern. Conf. on High energy physics (Singapore, 1990), eds. K.K. Phua and Y. Yamaguchi, Vol. I (World Scientific, Singapore, 1991) p. 144. [ 5 ] K. Sliwa, Search for the top quark at the Fermilab Collider, in: Proc. Fourth heavy flavor Syrup. (Orsay, France, June 1991 ), in press. [6] R.H. Dalitz and G.R. Goldstein, Phys. Rev. D 45 (1992) 1531. [7] R.P. Feynman, Phys. Rev. Lett. 39 (1969) 1415. [8] J.H. Kuhn, Nucl. Phys. B 237 (1984) 77. [9 ] A. Czarnecki, M. Jezabek and J.H. Kuhn, Nucl. Phys. B 351 (1991) 70. [ 10] CDF Collab., F. Abe et al., Phys. Rev. Lett. 64 (1990) 142.